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We first construct indices for shares in five European sectors (banking, chemicals, insurance, oil and retail) and estimate the fair prices that a (currently unavaible) futures contract on each one of these indices would have had if it existed. We then show that hedging the sectoral exposure of a diversified portfolio of pan-European shares in the same sector with those futures contracts provides a superior risk-cover than an alternative hedging strategy using futures contracts based on either a cocktail of domestic stock market indices or a pan-European stock market index. We conclude that the introduction of futures contracts on pan-European sectoral indices should increase the efficiency of both the cash and futures European equity markets